Is it always true that $$\lim_{k \rightarrow \infty}\int_E |f-f_k(x)|d\lambda=0 ?$$

I see the striking similarity to Lebesgue's dominated convergence theorem, if one could use $\displaystyle\lim_{k \to \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$ to find some majorant $g$ for our $f_k$ it would be true, especially it would be true when the $f_k$ converge against $f$ from below.